Measure and

John C. Oxtoby

Measure and Category

A Survey of the Analogies between Topological and Measure Spaces

Second Edition

Springer-Verlag

N ew York Heidelberg Berlin

1. Measure and Category on the Line

The notions of measure and category are based on that of countability.

Cantor's theorem, which says that no interval of real numbers is countable, provides a natural starting point for the study of both measure and category. Let us recall that a set is called denumerable if its elements can be put in one-to-one correspondence with the natural numbers 1,2, ....

A countable set is one that is either finite or denumerable. The set of rational numbers is denumerable, because for each positive integer k there are only a finite number ^(~2k - 1) of rational numbers p/q in reduced form (q > 0, p and q relatively prime) for which

Ipi

+ q

=

k. By numbering those for which k = 1, then those for which k = 2, and so on, we obtain a sequence in which each rational number appears once and only once. Cantor's theorem reads as follows.

Theorem 1.1 (Cantor). For any sequence {an} of real numbers and for any interval I there exists a point p in I such that p =1= an for every n.

One proof runs as follows. Let 11 be a closed subinterval of I such that a1 ¢ I l' Let 12 be a closed subinterval of 11 such that a2 ¢ 12, Proceeding inductively, let In be a closed subinterval of I n-1 such that an ¢ In. The nested sequence of closed intervals In has a non-empty intersection. If pEn In> then pEl and p =1= an for every n.

This proof involves infinitely many unspecified choices. To avoid this objection the intervals must be chosen according to some definite rule.

One such rule is this: divide In _ 1 into three subintervals of equal length and take for In the first one of these that does not contain an' If we take 10 to be the closed interval concentric with I and half as long, say, then all the choices are specified, and we have a well defined function of (l, a1 , a2 , ... ) whose value is a point of I different from all the an'

The fact that no interval is countable is an immediate corollary of Cantor's theorem.

With only a few changes, the above proof becomes a proof of the Baire category theorem for the line. Before we can formulate this theorem we need some definitions. A set A is dense in the interval I if A has a non- empty intersection with every subinterval of I; it is called dense if it is

dense in the line R. A set A is nowhere dense if it is dense in no interval, that is, if every interval has a subinterval contained in the complement of A. A nowhere dense set may be characterized as one that is "full of holes."

The definition can be stated in two other useful ways: A is nowhere dense ifand only if its complement A' contains a dense open set, and if and only if

A

(or A -, the closure of A) has no interior points. The class of nowhere dense sets is closed under certain operations, namely

Theorem 1.2. Any subset of a nowhere dense set is nowhere dense.

The union of two (or any finite number) of nowhere dense sets is nowhere dense. The closure of a nowhere dense set is nowhere dense.

Proof. The first statement is obvious. To prove the second, note that if A 1 and A2 are nowhere dense, then for each interval I there is an interval 11 C I - Al and an interval 12 C 11 - A 2. Hence 12 C I - (AI uA2 )·

This shows that Al uA2 is nowhere dense. Finally, any open interval contained in A' is also contained in A -'.

0

A denumerable union of nowhere dense sets is not in general nowhere dense, it may even be dense. For instance, the set of rational numbers is dense, but it is also a denumerable union of singletons (sets having just one element), and singletons are nowhere dense in R.

A set is said to be of first category ifit can be represented as a countable union of nowhere dense sets. A subset of R that cannot be so represented is said to be of second category. These definitions were formulated in 1899 by R. Baire [18, p. 48], to whom the following theorem is due.

Theorem 1.3 (Baire). The complement of any set of first category on the line is dense. No interval in R is of first category. The intersection of any sequence of dense open sets is dense.

Proof. The three statements are essentially equivalent. To prove the first, let A =

U

^An be a representation of A as a countable union of nowhere dense sets. For any interval I, let 11 be a closed subinterval of I -AI. Let 12 be a closed subinterval of 11 -A2' and so on. Then nIn is a non-empty subset of I - A, hence A' is dense. To specify all the choices in advance, it suffices to arrange the (denumerable) class of closed intervals with rational endpoints into a sequence, take 10

=

I, and for n > 0 take Into be the first term of the sequence that is contained in In-I-An·

The second statement is an immediate corollary of the first. The third statement follows from the first by complementation.

0

Evidently Baire's theorem implies Cantor's. Its proof is similar, although a different rule for choosing In was needed.

2

Theorem 1.4. Any subset of a set of first category is of first category.

The union of any countable family of first category sets is of first category.

It is obvious that the class of first category sets has these closure pro- perties. However, the closure of a set of first category is not in general of first category. In fact, the closure of a linear set A is of first category if and only if A is nowhere dense.

A class of sets that contains countable unions and arbitrary subsets of its members is called a a-ideal. The class of sets of first category and the class of countable sets are two examples of a-ideals of subsets of the line. Another example is the class of nullsets, which we shall now define.

The length of any interval I is denoted by

III.

A set A C R is called a nullset (or a set of measure zero) if for each 8 > 0 there exists a sequence of intervals In such that A cUI nand 2: II nl < 8.

It is obvious that singletons are nullsets and that any subset of a nullset is a nullset. Any countable union of nullsets is also a null set. For suppose Ai is a nullset for i = 1,2, .... Then for each i there is a sequence of intervals Iij (j = 1, 2, ... ) such that Ai

c UJij

and 2:jllijl < 8/2i. The set of all the intervals Iij covers A, and 2:ijllijl < 8, hence A is a nullset.

This shows that the class of null sets is a a-ideal. Like the class of sets of first category, it includes all countable sets.

Theorem 1.5 (Borel). If a finite or infinite sequence of intervals In covers an interval I, then 2: II nl ~ Ill.

Proof. Assume first that I / = [a, b] is closed and that all of the intervals In are open. Let (a1 , b 1) be the first interval that contains a. If b 1 ~ b, let(a2 , b2 } be the first interval of the sequence that contains b1 • If bn-1 ~ b, let (an, bn} be the first interval that contains bn-^{1 •} This procedure must terminate with some bN > b. Otherwise the increasing sequence {bn}

would converge to a limit x ~ b, and x would belong to

h

for some k.

All but a finite number of the intervals (an, bn) would have to precede

h

in the given sequence, namely, all those for which bn - 1 Elk' This is impossible, since no two of these intervals are equal. (Incidentally, this reasoning reproduces Borel's own proof of the "Heine-Borel theorem"

[5, p. 228].) We have

and so the theorem is true in this case.

In the general case, for any a > 1 let J be a closed subinterval of I with

IJI

=

III/a,

and let In be an open interval containing In with IJnl = allnl.

Then ^J is covered by the sequence {In}. We have already shown that

IIJnl

~

IJI.

Hence

aIllnl

=

IIJnl

~

IJI

=

III/a.

Letting

a-..1

we obtain the desired conclusion. 0

This theorem implies that no interval is a nullset; it therefore provides still another proof of Cantor's theorem.

Every countable set is of first category and of measure zero. Some uncountable sets also belong to both classes. The simplest example is the Cantor set C, which consists of all numbers in the interval [0, IJ that admit a ternary development in which the digit 1 does not appear. It can be constructed by deleting the open middle third of the interval [0, IJ, then deleting the open middle thirds of each of the intervals [0,1/3] and [2/3, 1], and so on. If Fn denotes the union of the 2ⁿ closed intervals of length 1/3ⁿ which remain at the n-th stage, then C =

n

^Fn-C is closed, since it is an intersection of closed sets. C is nowhere dense, since Fn (and therefore C) contains no interval of length greater than 1/3n • The sum of the lengths of the intervals that compose Fn is (2/3)", which is less than

t: if n is taken sufficiently large. Hence C is a nullset. Finally, each number xin(O, 1] hasa unique non-terminating binary development x = .X I X 2X3 ....

If Yi = 2xj, then .Y1Y2Y3 ... is the ternary development with Yi =l= 1 of some point Y of C. This correspondence between x and y, extended by mapping

°

onto itself, defines a one-to-one map of [0, 1] onto a (proper) subset of C.

It follows that C is uncountable; it has cardinality c (the power of the continuum).

The sets of measure zero and the sets of first category constitute two a-ideals, each of which properly contains the class of countable sets.

Their properties suggest that a set belonging to either class is "small"

in one sense or another. A nowhere dense set is small in the intuitive geometric sense of being perforated with holes, and a set of first category can be "approximated" by such a set. A set of first category mayor may not have any holes, but it always has a dense set of gaps. No interval can be represented as the union of a sequence of such sets. On the other hand, a nullset is small in the metric sense that it can be covered by a sequence of intervals of arbitrarily small total length. If a point is chosen at random in an interval in such a way that the probability of its belonging to any subinterval

J

is proportional to

IJI,

then the probability of its belonging to any given nullset is zero. It is natural to ask whether these notions of smallness are related. Does either class include the other?

That neither class does, and that in some cases the two notions may be diametrically opposed, is shown by the following

Theorem 1.6. The line can be decomposed into two complementary sets A and B such that A is of first category and B is of measure zero.

Proof. Let ai' a2 , .•. be an enumeration of the set of rational numbers (or of any countable dense subset of the line). Let Ijj be the open interval

4

with center ai and length 1/2ⁱ+j • Let Gj=

Uf;,

¹

Iij

(j= 1,2, ... ) and B =

ni=

^{1 Gj • For any B} > 0 we can choose} so that 1/2^j < B. Then B C

UJij

and

Li IIijl

=

Li

^{1/2; + j} = 1/2

j

_< B. Hence B is a nullset. On the other hand, Gj is a dense open subset of R, since it is the union of a sequence of open intervals and it includes all rational points. Therefore its complement Gj is nowhere dense, and A = ^{B' =}

U

^j Gj is of first category.

0

Corollary 1.7. Every subset of the line can be represented as the union of a nullset and a set of first category.

There is of course nothing paradoxical in the fact that a set that is small in one sense may be large in some other sense.